Spontaneous electric-polarization topology in confined ferroelectric nematics

Topological textures have fascinated people in different areas of physics and technologies. However, the observations are limited in magnetic and solid-state ferroelectric systems. Ferroelectric nematic is the first liquid-state ferroelectric that would carry many possibilities of spatially-distributed polarization fields. Contrary to traditional magnetic or crystalline systems, anisotropic liquid crystal interactions can compete with the polarization counterparts, thereby setting a challenge in understating their interplays and the resultant topologies. Here, we discover chiral polarization meron-like structures, which appear during the emergence and growth of quasi-2D ferroelectric nematic domains. The chirality can emerge spontaneously in polar textures and can be additionally biased by introducing chiral dopants. Such micrometre-scale polarization textures are the modified electric variants of the magnetic merons. Both experimental and an extended mean-field modelling reveal that the polarization strength plays a dedicated role in determining polarization topology, providing a guide for exploring diverse polar textures in strongly-polarized liquid crystals.


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Supplementary Fig. 11. State diagram dependent on the splay modulus K11 and polarization strength . We assume that the twist elastic modulus is equal to the bend elastic modulus, = = 2 pN. In the range of < 2.5 × 10 C m -2 , as decreases, the structure with the lowest energy changes from C-meron to escaped vortex structure, and then to bipolar structure. The reason for these transformations is that the above three structures contain more splay deformation in turn.
However, in the region of larger polarity, > 2.5 × 10 C m -2 , the bipolar structure with two +1 point defects at its two poles will consume huge energy due to the sharp increase of dipolar interactions, so C-meron is again the most favorable even in the range of small K11.

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Supplementary Fig. 12. State diagram dependent on the twist modulus K22 and polarization strength . We assume that the splay elastic modulus is equal to the bend elastic modulus, = = 2 pN. In the range of < 3.0 × 10 C m -2 , as decreases, the structure with the lowest energy changes from C-meron to escaped vortex structure, since the very small twist elastic modulus breaks the symmetry of the system and thus spontaneously forms chiral structures in the nonchiral system. However, this effect is suppressed in the high polarity region, i.e. = 3.0 × 10 C m -2 , since the escaped vortex structure contains large energy from dipolar interaction near the -1 point defect in its center.
Supplementary Fig. 13. State diagram dependent on the bend modulus and polarization strength . We assume that the splay elastic modulus is equal to the twist elastic modulus, = = 2 pN. As the bend modulus decreases, the structure with the lowest energy changes from C-meron to escaped vortex structure, and then to simple polarization vortex structure. The escaped vortex structure here is evolved from the simple polarization vortex structure which has a lot of bend deformation and has a defect line in its core. In order to reduce the total energy of structure, the LC molecules near the simple polarization vortex structure's core will escape along the z direction so that the line defect will degenerate into a point defect. The polarity affects the phase area of the three structures (high polarity shrinks the phase diagram area of the escaped vortex structure and increases that of the other two structures).

Materials
All commercial chemicals and solvents were used as received, unless stated otherwise.   19, 163.68, 162.37, 159.06, 153.91, 151.53, 133.45, 132.39, 120.25, 118.49, 115.51, 113.57, 110.39, 107.77, 65.01, 57.23, 14.96.  162.67, 155.94, 155.64, 150.87, 145.38, 133.78, 132.57, 125.30, 122.77, 117.97, 117.31, 116.02, 113.28, 109.84, 106.85, 65.15, 56.42, 14.63. 1g-j). The transition process suggests the metastability of the system. At low temperatures on cooling, though crystal is the most stable state, the slow nucleation and growth kinetics does not allow the system crystalized at the observation time scale (up to days). However, we can observe very small non-growing nucleus of crystal phase. Upon heating, the thermal perturbation provides fluctuation to the system to transit to the most-stable crystal phase. As shown in Supplementary Figs. 1f-g, the domains readily crystallized upon heating can be attributed to the seeding effect from the remaining nucleus. This is also clearly observed in the DSC measurement ( Supplementary Fig. 1).
The temperature of the Iso-NF transition is 64.6 °C and it exceeds the temperature limit of the oil immersion objective. Therefore, to obtain high resolution FCPM images, we choose an alternative material that shows the same NF droplets but at low temperatures.
We synthesized two pure and stable conformers of DIO: trans-and cis-conformers, and mixed the cis conformer into the trans conformer.  Fig. 5). This confirms the general incidence of the electric polarization meron-like structures. The dark region of the FCPM images of the line disclination ( Supplementary Fig. 6a) is apparently extremely sharp and thin even when one deliberately uses a low-NA objective.

Supplementary
It is not consistent with the broaden black cylinder in NF droplets ( Supplementary Fig.   6b), suggesting that the disclination line is absent and the defects in the NF droplets.
Therefore, at this stage, the concentric director fields might be either meron-like structure or escaped vortex structure.
To further differentiate the topology, we simulate the cross-sectional fluorescence profiles of these three structures and calculate the fluorescence intensity along paths in the center area and in the non-center areas ( Supplementary Figs. 7a-c). The same profiling treatments are employed to the experimental FCPM images to obtain the signal profile ( Supplementary Fig. 7d). From the signal profile of the escaped vortex ( Supplementary  Fig. 7c), it is seen that, the signal profile along the center path is the thinnest, and outsides exhibit symmetrically identical signal distribution with respect to the center line. It is because the polarizations in the center line of the vortex lie in-plane without severe tilting.
Predicted from the simulated meron-like structure , as the singularity is missing in the droplet center, the decrease of the intensity in the droplet center is due to the out-of-plane tilting of the polarizations. Since the NF-glass interface poses a degenerate anchoring, if the directors are closer to the substrates, the directors therefore exhibit less out-of-plane tilting so the fluorescence intensity decreases will be less in the area closer to the substrates than in the midplane. This feature is represented in the fluorescence cross section profile and signal profile of the meron-like structure: the extinction area in the droplet center is larger than that in the non-center area ( Supplementary Fig. 7b). Comparing the numerical FCPM images with the experimental results, it is clear that the NF droplets demonstrate the meron-like structure topology ( Supplementary Fig. 7d).
Supplementary Discussion 4. According to our observation, RM-OC2 directly transits from isotropic to NF with an intermittent co-existence (Figs. 2b,c). When the sample is further cooled down, the droplets merge and transit to a band-like texture, where the line disclinations run mainly along the rubbing direction (Fig. 2d). The corresponding director and polarization field can be directly visualized by SHG microscopy. Supplementary Fig.   3a demonstrates a large-area 2D SHG confocal polarizing microscopy (SHG-CPM) image of the band texture for the cell midplane during the cooling at 30 °C. In each domain, the SH signal shows the maximum intensity when the polarization is parallel to rubbing direction. This means the polarization points along the rubbing direction. Similar to Ref.